Questions tagged [rank-1-matrices]

Question 1

Given a fundamental $1$-form $$ g_{ij}= \delta_{ij} + \frac{\partial f}{\partial u_i}\frac{\partial f}{\partial u_j} $$ for $i, j = 1, 2,\dots, n$, the matrix of $[g]$ can be expressed as $$[g]=I_{n\...

Question 2

I'm trying to understand how the Howell normal form of a $1\times n$ matrix over the ring $R=\mathbb{Z}/\mu\mathbb{Z}$ looks like. If I undertand correctly, the only elementary operation allowed in ...

Question 3

I would like to know if anyone knows any reference about rank one update of Schur decomposition. Assume that we know the Schur decomposition of $A,$ which is $A=QUQ^{-1},$ where $Q$ is unitary and $U$ ...

Question 4

Singular value decomposition of a matrix $A$ of dimension $m \times n$ reads $A=U\Sigma V^+$. It can be seen as a way to decompose the matrix into a sum of rank-1 matrices: $$A=\sum_{i=1}^r \lambda_i ...

Question 5

I have a tall rank-$1$ matrix ${\bf Y} \in {\Bbb C}^{n \times m}$ (where $n > m$) whose singular value decomposition (SVD) is $$ {\bf Y} = \sigma_1 {\bf u} {\bf v}^\ast. $$ I add another $n \times ...

Question 6

The problem starts from my considerations in the answer If $ A^2 + AB + B^2 = 2BA, $ prove that $ AB = BA = O_2$, where I applied the matrix determinant lemma. Evidently, for any invertible matrix $A$,...

Question 7

I am interested in the eigenvalues of some square $n \times n$ matrix $A = B \circ ( u x^T)$, where $u = (1, 1, \ldots, 1)$ the vector of all ones, and $x$ and $B$ a random vector and random matrix ...

Question 8

Let $\bf A$ be an invertible $m \times m$ real matrix. Is there any simplified expression for the following $m^2 \times m^2$ rank-$1$ matrix? $$\operatorname{vec}({\bf A}) \operatorname{vec}^\top \...

Question 9

I am studying sparse partial least-squares (SPLS) regression, and I am interested in the mathematical foundations behind this method. The algorithm is proposed by Kim-Anh Lê Cao et al.$^\color{magenta}...

Question 10

Let $M$ be an $m\times n$ matrix of rank $r$. I am interested to express $M$ as $C_1 + \ldots + C_r$ where each $C_i$ is a rank-1 matrix. How many $C_i$'s of rank-1 am I allowed to fix (if any) and ...

Question 11

Let $I$ be the identity matrix and $P$ be an irreducible $n$ by $n$ row stochastic matrix. Let $d$ be a stochastic (column) vector and $e$ be an all one (column) vector. Let $t > 0$ be a real ...

Question 12

Question: I need help to prove the following statement. Let $W_i:=w_iw_i^T\in\mathbb{R}^{n\times n}$, for $n$ even, be symmetric rank-1 matrices, $J=-J^T$ the canonical symplectic matrix and define ...

Question 13

Suppose the exact hessian $H^\star$ as function of vector x (no need to further be specified) and the initial SR-1 approximation $H$ are globally bounded in some norm of your choice by some real ...

Question 14

For a fixed vector $\boldsymbol{v}\in\mathbb{R}^d$, I have the matrix $M=\boldsymbol{v}\boldsymbol{v}^\top$, from which I need to recover $\boldsymbol{v}$ (up to sign). I know that $\boldsymbol{v}$ is ...

Question 15

Given a square matrix $\bf A$ and two vectors $\bf u$ and $\bf v$ such that ${\bf 1}^\top {\bf u} = 1$ and ${\bf v} = {\bf 1}$. Is there some relationship between the singular values of $\bf A$ and ...

Stack Exchange Network

Questions tagged [rank-1-matrices]

Computing the determinant of the matrix of the fundamental $1$-form

Howell normal form of a $1\times n$ matrix

Efficient way for finding rank-one update of Schur decomposition?

Interpretation of matrix product $AB$ as sum of matrices of rank $1$

Perturbation of singular values of rank-$1$ matrix

Implications of formula for matrix determinant lemma

Eigenvalues of the Hadamard product of a random matrix and a rank-one matrix

Simplifying $\operatorname{vec}({\bf A}) \operatorname{vec}^\top \left({\bf A}^{-1}\right)$

Sparse least-squares regression and SVD

Completing a rank-1 decomposition of a matrix

Stability of rank-$1$ perturbation of stochastic matrix

Showing existence of symplectic transformations preserving a quadratic form

bound for SR1 update

Computing eigenvalues of a rank 1 matrix

How does a rank-$1$ update affect the singular values?

Hot Network Questions